ADDING DEPTH TO ILLUSIONS

Older computer screens don't have have precisely linear geometry.
The stereo illustrations look best when viewed on an LCD screen,
or from a printed copy.
I have used 16 color GIFs to minimize loading time.
These were derived from source files created with DesignCad,
and interested persons can request
source files in that format, or in DXF format. They have been converted
to bitmaps for presentation here.

Disclaimer: Some of the isometric illusions below
were creations of Swedish Artist Oscar Reutersvärd. My specific
stereo interpretations of them are not to be blamed on him.
I am not the first to do stereo versions of these. I would welcome any
documented references on the history of these illusions. Those
that are my original creations are so indicated.

ILLUSIONS WITH PERSPECTIVE

Since many of the common illusions seem to depend on "false" perspective,
or on the lack of true stereoscopic depth, we might inquire how illusions
could be constructed with true perspective, and possibly even in full stereo
depth.
Conventional wisdom holds that such illusions are destroyed when depicted
with accurate stereoscopic depth.

The ultimate challenge would be to create illusory sculpture. Some illusions
can be realized as sculptures, but must be viewed from one particular point
with just one eye. Such is the case with the Necker cube, above, sometimes
drawn as shown above right, and called the "crazy crate."

We will begin by considering whether some of the common illusions could be
altered to include perspective.

ISOMETRIC ILLUSIONS

Many classic "tribar" illusions are conventionally drawn in isometric
fashion,
in which parallel lines are rendered parallel on the page, there being no
convergence toward a vanishing point. Objects of the same size
are also the same size on the drawing surface no matter how far away they are.
Engineering drawings are often isometric to make it easy to preserve relative
lengths no matter what the inclination of a line to the observer's
line of sight. Isometric drawings do not preserve angles.
Any angle of 90° is rendered as 120°. Perspective drawing doesn't
preserve angles either.
Cartesian coordinate systems are often depicted isometrically,
with one axis vertical, and the other two axes making
angles of 120° with it.

Isometric drawing is essential for some of these illusions.
The ambiguous staircase illusion would lose its illusory character if
drawn in true perspective. But a few of the isometric illusions can be
successfully rendered in perspective, as I will demonstrate.

In fact the prototype of them all, the "Penrose"
illusion, was first presented with a distinct
suggestion of perspective.[1] We show it that way
here. Each bar has convergence toward a
vanishing point. The original drawing, in the
Penrose paper, was shown with some shading as
well.

Isometric illusions depend upon two deceptions.

The false perspective
in which even supposedly receding parallel lines remain parallel.
There's no size change with distance.

Ambiguous or impossible
connection between picture elements.
Picture elements are connected, even though perspective
cues tell us they should not be near each other.

The example below left is shown in isometric style. The conventional
wisdom has been that perspective rendering of such
pictures is not possible, or would destroy the illusion.

The same illusion is shown at the right in perspective, using two
vanishing points. The illusory nature of the object is certainly not
diminished by this presentation, though I don't claim that it
is greatly enhanced.

Some persons experience an interesting effect while comparing
these two pictures. Look at the perspective view for a while,
then shift your attention to the isometric view at the right.
The isometric view may now appear "wrong" or "warped", and you may feel
that the top and bottom of the frame are not parallel, and diverge at
the rear of the frame!

STEREO DRAWING

Stereo drawings require two pictures, one for each eye. To view such
drawings requires some practice. Generally two methods are used: (1) parallel
viewing and (2) cross-eyed viewing. Both methods require one to learn a new
visual skill.

Normally when we look at the "real" world, our eyes converge on an object and
they also focus on the same object. We habitually do this, and our brains
have become accustomed to a one-to-one correspondence between focus and
convergence.

We can learn to "unlock" focus and convergence, enabling us to view
stereo pairs without optical aid.
In this document we use a display method which can be adapted to either
parallel-eye viewing or crossed-eye viewing. Three drawings will be
shown side by side. The middle drawing is to be viewed with
the right eye. The other two are identical and one is to be viewed with
the left eye. Here's how to view them.

(1) Parallel viewing. This is sometimes called wall-eyed viewing.
Use the left and middle pictures only. Look at a
distant object then bring your eyes down to the paper trying not to
converge or focus on the paper. You'll see a blurred double image.
Consciously try to bring the double image into one. Now try to focus
your eyes on it without allowing the two images to drift apart.

Parallel viewing is limited to pictures about 2.5 inches wide, the
spacing of the typical human eye.

(2) Cross eyed viewing. Use the right and middle pictures only.
Hold your finger exactly halfway between your eyes and the page. Focus on
your finger. Your eyes will be converged on the finger also, and you
should be conscious of the two pictures out of focus behind your
finger, but probably nearly coincident. Move your finger a bit until
the two pictures are fully coincident. Now hold the convergence while
refocusing from your finger to the page. Finally, you can remove your
finger from the field of view.

Crossed eyed viewing can be used with large pictures.
Paintings have been presented this way. Salvadore Dali painted some
of this kind. For those who can learn the skill of viewing, this is
one of the most effective ways for viewing stereo without special
glasses.

(2) Mirror method. For completeness, we illustrate a method rarely used in printed books. Popular Photography magazine experimented with it for a while. One picture of the pair is printed normally, ad the other is printed beside it, but reversed left/right. That picture is viewed through a mirror which re-inverts it. We will not use that method here.

For practice, try this illustration from Sir Charles Wheatstone's book
The Stereoscope.[2]

Don't expect to succeed the first time. This skill takes conscious effort and
concentration. When you do succeed, you'll see the pictures snap into full
three-dimensional depth. The picture will look like a wire-frame box.
You'll actually see two 3-D images, one with normal depth, one with inverted
(pseudoscopic depth). On either side of these you'll see fainter, phantom
images with no depth. Ignore them.

Here's some more practice examples:

If you use cross eyed viewing on the pair intended for parallel, or
vice versa, you will see a "pseudoscopic" depth, in which near and far are
reversed. In the first picture, the pseudoscopic view appears as if you are
looking down onto a truncated pyramid. In the second picture, the cone seems
upright in the normal view, but tilted back and viewed from its base in
the pseudoscopic view. Wire-frame stereo drawings often look interesting
either way.

Here's another example for practice.

This coiled spring is more difficult to view:

Now, can illusion pictures be drawn this way? Some can. The three-tined
fork illusion, sometimes called "Schuster's conundrum," succeeds remarkably
well. This is strictly an illusion of ambiguous connectivity; there's no depth
ambiguity at all.

Here's my color 3d rendition of the classic "Crazy Crate".

Let's try the Penrose Illusion (impossible triangle).
Here we use the fact that a horizontal line
has ambiguous depth even in stereo. So we've oriented the triangle with one
side horizontal. The other sides have been given true stereoscopic depth,
but no perspective depth cues are used.

But now try viewing this version. Here we haven't used the cheap trick of
horizontal lines. We've used a different cheap trick. We've simply expanded
the horizontal dimension of one picture by about 5%.

Why should this work at all? It seems to defy logic. Let's try the same
trick with some other isometric pictures.

And another:

Feel free to view any of these pseudoscopically. It doesn't seem to matter
a lot. You get a vague sensation of stereoscopic depth either way!

Some people have a weak perception of depth in such drawings even when
both pictures
are identical! This may be due to the artificial method for viewing them,
particularly the slight keystone distortion of each picture when cross-eyed
viewing is used. The absence of focus cues may play a role also.

I haven't prejudiced you by suggesting what you should see in these
examples. Generally one experiences the same ambiguity of depth, as in the
"flat" isometric version, but there's an added cue of
stereoscopic depth as
well. The stereoscopic depth seems to fluctuate depending on where one
fixes one's attention within the picture. Clearly we are getting a conflict
of depth and solidity cues. The stereoscopic cues and the isometric
perspective cues do not agree.

ILLUSIONS OF SHAPE

There's a large class of illusions called pattern-dominance or pattern-conflict
illusions. They fall within a larger class of illusions of shape.

Pattern-dominance illusions, as usually presented, seem to be
strictly due to conflict of overlapping patterns in a single plane.
Our perception
of the geometry of one pattern is altered by the presence of the other
pattern. The illusion seems not to rely upon any suggestion of perspective
in the drawing.

Most people judge that the circles in the left drawing a bit off-round, being gently
flattened at four places, near the corners of the squares. Few would say
that the circles distort the squares in this case.

We can test this by making another drawing (on the right) in which the squares dominate
the background field of view, while a lone circle competes with that.
Will the circle show distortion, but the squares remain square? Yes, that's what most people see.

Both versions of the illusion persist in stereo rendering even though the circles and squares now lie in different planes when seen in depth.

Some explain the following illusion by claiming the radial lines
are interpreted by the brain as parallel lines receding to a vanishing
point. This supposedly makes one circle (usually the right one) seem
smaller, though they are drawn the same size and therefore subtend the
same angle to the eye. Again, I find this explanation unpersuasive.

The Ehrenfels illusion presents a perfect square upon a background
of radial lines. The square seems tilted forward. (Or, it appears to be
a rhombus, with the top edge longer than the lower edge.) It still seems
tilted or distorted
when the square is drawn on a transparent sheet held some distance in
front of the plane of the radial line pattern. In stereo rendering
there's a strong illusion that the square is tilted, the top edge nearer
than the lower edge, even though there are no stereoscopic cues to
support this interpretation.

A related illusion, the Herring illusion, presents parallel lines against
a background of radial lines. The parallel lines appear bowed or bent.
They still appear bent if they are on a transparent sheet some distance in
front of the plane of the radial line pattern. This fact comes through
in stereo rendering also.

This one is repeated below in larger scale, for crossed-eye viewing only.

Some argue that the pattern of radial lines is suggestive of
perspectiveof parallel lines receding to a vanishing point. I don't
find that explanation persuasive. But it's hard to devise pattern
conflict illusions in which one or the other of the patterns can't
be interpreted as having some characteristic of perspective.

If you are viewing this illusion in stereo from the monitor screen you may
see that one of the parallel lines seems nearer than the other. If you
view it from the printed page, they seem to be at the same distance. This
is due to horizontal non-linearity of the monitor's display. Also, if you have
uncorrected astigmatism in one or both eyes you may notice that the radial lines
do not appear equally
distinct. This is similar to the standard astigmatism test chart, which also has a
radial pattern of lines.

MORE 3D ILLUSIONS

[April, 2002] I finally got around to rendering my gear illusions
in stereo.

As with most illusions, it uses several forms of deception. The following picture shows in the ovals, the central illusion, which is nothing more than Mach's "open book" illusion, shown to the right. It can be seen as facing pages of an open
book, or as the front and back cover of a an open book seen from its back. This simple isometric illusion is the basis of the ambiguous staircase illusion as well as many others.

The 2D version of course came first. This raised the question "Could this be rendered in 3D". The drawing was isometric, which has no classical perspective (no vanishing points, which give apparent depth to flat pictures). But you can still employ stereo parallax in isomemtric drawings, for the stereo disparity overrides weaker depth clues. That's easily accomplished with ellipses (and gears) by altering their width to height ratio. The two gears of the left eye picture are fattened horizontally, and the larger gear of the lelft eye picture is made narower. This is done only on either side of the vertical line passing trhough the illusory meshed gear teeth. In the CAD drawing, this breaks some of the points where lines join, and these must be repaired by hand. All this surgery was done only on the left eye picture. Then when all was fixed in the CAD drawing, it was converted to a GIF, and colorized with a paint program. The colors help to emphasize the illusion, but must be done consisently. Notice that the left gears have teeth with orange tops, but the right gear has orange valleys between the teeth. This is necessary because in the ambigous regions where they teeth mesh, the orange top and valley are the same parallelogram. Similar constraints apply to yellow and blue faces. Coloring these can warp one's mind, leading to mistakes. If any mistakes remain, please let me know.

One might even say that the "colors transfer from one gear to the other" where the teeth mesh. Now if someone wants a real challenge, how about making an animated GIF of these gears rotating in 3d. (I can supply the original CAD structure in DXF format if anyone wants to try.)

A CHALLENGE

My ambiguous ring illusion (below) makes use of the inherent ambiguity of
circles and ellipses seen in perspective. Cover the left or right third
of the picture and everything seems conflict-free.

Here's another example of ellipse ambiguity. View this either cross-eyed or
parallel. One set of rings will seem perfectly normal, with the rings lying
in planes nearly perpendicular. But the other one, when examined carefully,
will soon begin to seem "wrong", and finally will seem to have an unnatural
twist where the rings link. Here's a conflict between stereo depth cues and
the drawing cues which tell us which part of the ring is "in front". This
is what happens when a magician messes up the Linking Rings trick.

References

[2] Wheatstone, Sir Charles. "Contributions to the Physiology of Vision. Part the First; On Some Remarkable, and Hitherto Unobserved, Phenomena of Binocular Vision," Philosophical Transactions of the Royal Society, 1838, Part 1, pp 371-94. Reprinted in The Scientific Papers of Sir Charles Wheatstone, London, 1879, pp. 225-259.
Online copy, complete.

[3] Seckel, Al. The Art of Optical Illusions. Carlton Books, 2000.

[4] Seckel, Al. More Optical Illusions. Carlton Books, 2002.

Al Seckel's books are, in my biased opinion,
the best general illusion collections published,
and are very reasonably priced. See these
descriptions, and order them
from your favorite book source.

This document is an ongoing project, for which feedback is
welcomed by the author, who hopes that these drawings can stimulate an exchange of ideas.
Use the address shown here.
Expect to see additions and changes in this section of my web pages
in the future.